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Hyperconnectedness IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 4 Ver. IV (Jul - Aug. 2015), PP 66-72 www.iosrjournals.org S*- Hyperconnectedness Adiya K. Hussein Department of Mathematics, College of Basic Education, Al- Mustansiriya University, Iraq Abstract: The aim of this paper is to introduce and study hyperconnectedness in supra topological spaces, which we called, supra hyperconnectedess and denoted by S* hyperconnectedness. Several characterizations of supra hyperconnectedness are provided. We give conditions under which supra hyperconnectedness is preserved. Further, we introduce and study strong forms of supra hyperconnected spaces and obtain some characterizations of hyperconnected spaces. Keywords: supra topology, supra open set, hyperconnectedness, hyperconnectedness, S* dense. I. Introduction And Preliminaries In 1983 A. S. Mashhour et al. [1] developed the notion of supra topological spaces and studied the notions of supra- continuity. Devi et al. [2] introduce and studied a class of sets and maps called supra - open sets and supra - continuous maps. Arockiarani [3] introduced the class of supra semi- open (resp. supra regular open) sets. In 2010, Sayed et al. [4] introduced supra b- open sets and supra b- continuity and studied some of its properties. In [5], the concepts of supra - open sets, supra - continuity were introduced and studied. Levine [6] called a topological spaces X as a D- space if every non-empty open set of X is dense in X. Sharma [7] has shown that D- space is equivalent to hyperconnected space due to Steen and Seebach [8]. Noiri [9, 10] studied hyperconnected spaces. Ekici [11] introduced and studied hyperconnectedness in generalized topological spaces. In this paper we introduce and study the concept of supra hyperconnectedness. Several characterizations of supra hyperconnected spaces are investigated and some preservation theorems are given. Further, we obtain some characterizations of hyperconnected topological spaces. Let us recall the following definitions and results which are useful in the sequel. Definition 1.1. Let (X,) be a topological space and A be a subset of X. The closure and the interior of A will be denoted by cl(A) and int(A). A set A is called - open[12] (resp. preopen[13], semi- open, [14], b- open[15], -open[16]) if A int(cl(int(A)) (resp. A int(cl(A)), A cl(int(A)), A int(cl(A)) cl(int(A)), A cl(int(cl(A))). The complement of the above sets are called there respective closed sets. The family of all - open ( resp. pre- open, semi-open, b- open, -open) sets are denoted by(X ) , PO(X), SO(X) ,BO(X) and O(X) respectively. Let P(X) be the power set of X, for a nonempty set X. Definition 1.2. [1] Let X be a nonempty set. A subset S* P(X) where P(X) is the power set of X, is called a supra topology if (1) X, S*. (2) S* is closed under arbitrary union. The pair (X, S*) is called a supra topological space. The members of S* are called supra –open sets and denoted s* open sets. The complement of supra- open set is the supra closed sets, denoted by s* closed Remark 1.3. [1] Let (X, ) be a topological space. Then, O(X), PO(X), SO(X), BO(X) and O(X) are supra topologies associated with . Definition 1.4. [1] Let (X, S*) be a supra topological space and A X. Then S* - interior and S* - closure of A in (X, S*) defined as {U :U A,U S*} and {F : A F, X F S*}respectively. The S* -interior * * and S -closure of A in (X, S ) are denoted asint * (A) and cl * (A) respectively. s s Theorem 1.5. [17] Let (X, S*) be a supra topological space and A X. Then (1) A . (2) = A if and only if A S*. (3) A . DOI: 10.9790/5728-11446672 www.iosrjournals.org 66 | Page Hyperconnectedness (4) =A if and only if A is supra closed. (5) x if and only if every supra open set Ux containing x, Ux A . Definition 1.6. A subset A of a supra topological space (X, S*) is said to be * (1) S - - open[2] (briefly * -open) if A int * ( cl * (int * (A))). s s s s * (2) S - pre- open (briefly p * open ) if A int * ( cl * (A)). s s s * (3) S - semi- open[3] (briefly s * open ) if A ( (A)). s S* * (4) S - b-open[4] (briefly b * open ) if A cl * ( (A)) ( (A)). s s * (5) S - -open[5] (briefly * open ) if A ( ( (A)). s * (6) S - regular- open [3] (briefly r * open ) if A= ( (A). s The set of all -open (resp. , , b * open , , ) sets is s * denoted by (X ) (resp. P*O(X ) , S*O(X ) , B*O(X ) , *O(X ) ), R*O(X ) ). It is clear that (X ) B*O(X ) and *(X ) The complements of the above mentioned sets are called their respective closed sets. II. Hyperconnected Spaces In this section we introduce and study the notion of supra hyperconnectedness. Several characterizations are given. Definition 2.1. A subset A of a supra topological space (X, S*) is said to be (1) s* dense if (A)= X. (2) s* nowhere dense if ( (A))= . Definition 2.2. A supra topological space (X, S*) is said to be (1) Hyperconnected (equivalently X is hyperconnected) if A is * for every subset s dense A of X. (2) Connected ( equivalently X is connected) if X cannot be written as the union of nonempty and * disjoint sets A and B in (X, S ). (3) Irreducible (equivalently X is irreducible) if A B for every nonempty subsets A s* open and B of (X, S*). Remark 2.3. From Definitions 2.2 we have the following relation. (X, S*) is hyperconnected (X, S*) is connected The above implication is not reversible as shown in the following example: Example 2.4. Let X= {a, b, c}, S* = {X, , {a}, {b}, {a, b}}. Then (X, S*) is connected but not hyperconnected. cl * (A) s Theorem 2.5. Let (X, S*) be a supra topological space. The following are equivalent: (1) (X, S*) is hyperconnected. (2) A is or dense, for every subset A of X. (3) A B , for every nonempty subsets A and B of X. DOI: 10.9790/5728-11446672 www.iosrjournals.org 67 | Page Hyperconnectedness Proof. (1) (2): Let (X, S*) be a hyperconnected and A be a subset of X and suppose that A is not dense. Then (X\ (A)) =X\ ( (A)) X. Since ( (A)) , so by (1), ( ( (A))) = X. Since ( ( (A))) = X (A). Then (A) = X. Hence A is . (2) (3): Suppose that A B= for some nonempty subsets A and B of X. Then p * open int * (A) B= and A is nots . Since As S*, so A ( (A)). Hence A is not s * open s dense. This is a contradiction. b * open cl * * (3) (1): Let A B sfor every nonempty s subsets A and B of X. Suppose that (X, S ) is not * open hyperconnected. Then there iss a nonempty subset U of X such that U is not in X, thus (U) X. Hence X \ (U) and U are disjoint nonempty subsets of X. This is a contradiction. Hence (X, S*) is hyperconnected. By Definition 2.2(3) and Theorem 2.5, we have, * Corollary 2.6. The supra topological space (X, S ) is hyperconnected if and only if it is irreducible. * * * Definition 2.7. The S - semi- closure( resp. S - pre- closure, S - -closure) of a subset A of a supra topological * s p space (X, S ), denoted by c * (A)( resp. c * (A), c * (A)) is the intersection of all s * closed ( resp. s s s s p * closed , * closed ) sets of X containing A. s s The following Lemma is needed in the next result. Lemma 2.8. Let A be any subset of a supra topological (X, S*). Then A ( ( (A))) = (A). Theorem 2.9. For a supra topological space (X, S*), the following are equivalent. (1) (X, S*) is hyperconnected. * (2) s Anowhere is for every subset A X. (3) A is for every subset A X. (4) A is for every subset A X. s* dense (5) (A)= X for every subset A X. (6) (A)= X for every subset A X. (7) (A) = X is for every subset A X. s* open Proof. (1) (2): Suppose that A is a nonempty subset of X. Hence ( (A))) . Then (A)= ( ( (A)))= X. (2) (3): Since every is , we have (3). (3) (4): Since every is , we have (4). (4) (5): Suppose that A is a set such that (A) X. Then there is a nonempty set U such that U A= . Hence (U) A= . Then, by (4), = (U) (A)= (U) which is a contradiction. DOI: 10.9790/5728-11446672 www.iosrjournals.org 68 | Page Hyperconnectedness (5) (6): Suppose that there is a nonempty set A such that (A) X. Hence there is a set U such that U A= . So U ( A) = . Hence, by (5), (A) = (U) (A) U (A)= . This is a contradiction. (1) (7): Let A be a non empty subset of X. So int * (A) , hence, by assumption, s ( (A))= X. Byp Lemma* open 2.8, (A)int * A ( ( (A))) = A (X)= X. s s s * open Hence (A) = X. s cl * (7) (6): This is obvious. s (6) (1): Let A be a nonempty subset of X. So A is . Hence by (6), (A)= X. * open s Since (A) (A). Then (A)= X. So we have (1).
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